Local Shannon entropy

When the probability of an event tends to unit or zero, the Shannon entropy is close or equal to zero. Thus, if probability pji of localization of amino acidj in position i equals unit (an absolutely certain event), then... 

In this work, we will also analyze, apart from C(LMC) and C(FS), the Cramer-Rao complexity C(CR), also as the product of a local and a global measure, keeping the first one as the Fisher information I, and replacing the Shannon entropy exponential by the variance V, giving rise to... 

Consider the Fisher information for locality [1,2], called intrinsic accuracy, which historically predates the Shannon entropy by about 25 years being proposed at about the same time that the final form of quantum mechanics was shaped. This local measure of the information content of the continuous (normalized) classical probability density p r) reads... 

The corresponding Sr(Sp) plot for the hydride-exchange Sn2 process again exhibits the maximum (minimum) at TS, with two additional minima (maxima) in its vicinity, where the bond breaking is supposed to occur. These additional, pre-and post-US features are symmetrically placed in the entrance and exit valleys, relative to TS structure, but now they appear at roughly the same values of the Intrinsic Reaction Coordinate (IRC) in both the position and momentum representations. This is in contrast to the two-stage abstraction mechanism, where in the entrance valley the p-space maximum of the Shannon entropy has preceded the associated minimum observed in the r-space. This simultaneous r-localization (p-delocalization) may be indicative of the single-St p mechanism in which the approach of the nucleophile is perfectly synchronized with the concomitant... 

Chemistry, Abstract No. 267 (2005). Shannon Entropy as a Local Surface Property. 

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